$ D = \left[\begin{array}{rr}4 & -2 \\ 4 & 1 \\ 1 & 2\end{array}\right]$ $ E = \left[\begin{array}{rrr}-1 & 0 & 2 \\ -1 & -2 & 0 \\ -1 & 0 & 2\end{array}\right]$ Is $ D+ E$ defined?
Explanation: In order for addition of two matrices to be defined, the matrices must have the same dimensions. If $ D$ is of dimension $( m \times  n)$ and $ E$ is of dimension $( p \times  q)$ , then for their sum to be defined: 1. $ m$ (number of rows in $ D$ ) must equal $ p$ (number of rows in $ E$ ) and 2. $ n$ (number of columns in $ D$ ) must equal $ q$ (number of columns in $ E$ Do $ D$ and $ E$ have the same number of rows? Yes Yes No Yes Do $ D$ and $ E$ have the same number of columns? No Yes No No Since $ D$ has different dimensions $(3\times2)$ from $ E$ $(3\times3)$, $ D+ E$ is not defined.